The Problem: Find two linear functions f(x) and g(x) such that their product h(x) = f(x).g(x) is tangent to each of f(x) and g(x) at two distinct points.
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I will define f and g as cotangent lines:
Definition: Two distinct linear functions, f and g, are cotangent if their graphs are tangent to the parabola that is the graph of f.g.
To determine the nature of cotangent lines, I used an algebraic approach. First I defined f, g, and f.g as follows:
f(x) = ax + b
g(x) = cx + d
fg(x) = (ax + b)(cx + d)
Next , I set f = fg to find the values of x for which the first line intersects the parabola.
ax + b = (ax + b)(cx+d)
0 = (ax + b)(cx+d) - (ax +b)
0 = (ax + b)(cx + d - 1)
ax + b = 0 or cx + d - 1 = 0
This leads to the solutions:
If the line is tangent to the parabola, there is only one point of intersection. By setting the two values of x equal to each other, an equation is formed relating the values of a, b, c, and d.
-bc = a - da
a = da - bc
By a similar process, setting cx + d = (ax + b)(cx + d) will yield the equation:
c = bc - da
This yields a very important fact: a = -c. We can conclude the following:
Theorem: The slopes of two cotangent lines are opposites.
Substituting -a for c in either equation yields another important fact:
a = da - b(-a)
a = da + ba
a = a(d + b)
1 = d + b
Thm: The sum of the y-intercepts of two cotangent lines is 1.
Now, given any linear function, it is quite simple to find its the function for its cotangent line. Try this example:
What is the function for the line cotangent to f(x) = 5x - 4?
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Calculus: Use the first derivative of fg to verify the two theorems stated above.
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Cotangent Parabolas: Is there a second parabola that is tangent to two cotangent lines?
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EMAT 6680 Class Page for Kyle Schultz